Order Out of Chaos in the Three-Body Problem: Regions of Escape

Abstract

The dynamical behavior of the general three-body problem with equal masses is studied in the regions where the motion leads to triple close approaches (e.g., near equilateral triangular position with zero initial velocities). Due to the strong third body effect, the classical perturbation approach is no longer applicable in these regions. This paper consists of two parts. In the first part a qualitative analysis of a simpler case, namely the planar isosceles three-body problem with negative energy is offered. Various aspects of this problem have been investigated by many authors in the recent years. In this paper, it is shown that a necessary and sufficient condition for escape is a syzygy crossing immediately after a binary collision. Furthermore, we have shown that the initial conditions leading to escape form open sets (i.e., connected regions) in the phase space. On each set the number of syzygy crossings as well as the number of binary collisions between consecutive syzygy crossings are invariant. In the second part the results are confirmed by finding some of these regions, using numerical integration. In each region the parameters of escape (i.e., orbital elements of the binary, velocity of the escaping body, etc.) are continuous functions of the initial conditions, and the boundary provides the measure for dynamical sensitivity of initial conditions. A comparison of the measure for dynamical sensitivity with a measure for numerical sensitivity (i.e. time reversal test) enable us to give a precise definition for computability. According to our numerical results, the higher number of syzygy crossings leads to higher dynamical sensitivity (i.e., smaller regions). This suggests the non-computability of the escape solutions with very high number of syzygy crossings, and the difficulty of detecting the regions of escape in these cases. Finally, using numerical integration, the results are extended to a more general case where the isosceles symmetry is broken and a special set of initial conditions for the planar three-body problem is considered.KeywordsInitial PositionBinary CollisionTriple CollisionDynamical SensitivitySpecial Initial ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.